Answer-first summary for fast verification
Answer: 2.003
## Explanation Expected Shortfall (ES), also known as Conditional VaR, measures the average loss in the tail beyond the VaR level. For a 95% confidence level, ES is calculated as the average of VaR values from 95% to 100%. ### Calculation: - For discrete data points, we take the average of VaR values from 95% to 100% - Given VaR values: - 95%: 1.6392 - 96%: 1.7507 - 97%: 1.8808 - 98%: 2.0537 - 99%: 2.3263 ### Expected Shortfall at 95%: \[ ES_{95\%} = \frac{1.6392 + 1.7507 + 1.8808 + 2.0537 + 2.3263}{5} \] \[ ES_{95\%} = \frac{9.6507}{5} = 1.93014 \] However, this gives us 1.930, which matches option B. But let me verify the calculation more carefully. Actually, for Expected Shortfall at 95% confidence level, we need to average the VaR values from 95% to 100%: \[ ES_{95\%} = \frac{1.6392 + 1.7507 + 1.8808 + 2.0537 + 2.3263}{5} = \frac{9.6507}{5} = 1.93014 \] This equals approximately 1.930, which corresponds to option B. **Wait, let me recalculate:** \[ 1.6392 + 1.7507 = 3.3899 \] \[ 3.3899 + 1.8808 = 5.2707 \] \[ 5.2707 + 2.0537 = 7.3244 \] \[ 7.3244 + 2.3263 = 9.6507 \] \[ 9.6507 ÷ 5 = 1.93014 \] Yes, this confirms the answer is **1.930** (Option B).
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Value at risk (VaR) determines the maximum value we can lose for a given confidence level. For this reason, Kenneth Fulton is concerned that the VaR is not providing the magnitude of the actual loss. He has prepared the following table based on the assumption that returns are normally distributed and a corresponding n = 5. What is the expected shortfall using the information in the following table at the 95% confidence level?
| Confidence level | VaR |
|---|---|
| 95% | 1.6392 |
| 96% | 1.7507 |
| 97% | 1.8808 |
| 98% | 2.0537 |
| 99% | 2.3263 |
A
0.687
B
1.930
C
2.003
D
2.054
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